Chapter 3 - MPC Control for ISWEC
Figure 3.27: Simulink Model - MPC with Known Disturbance
Chapter 3 - MPC Control for ISWEC
In the Simulink model in figure 3.27 it is possible to notice that the subsystemWAVES provides two signals. One of these is τw, who is the induced torque already discussed in the previous MPC versions, and Tau w Future Samples which is the new signal that implements the sequence defined in (3.30). The signal Tau w Future Samples is therefore fed to the MPC controller that will use the contained measures for predicting in accurate way the evolution of the system dynamics.
The figure 3.28 illustrates the subsystemCONTROLLERof the Simulink model in figure 3.27.
Figure 3.28: Controller Subsystem - MPC with Known Disturbance
It is worth to notice that the variable called Tau w Future Samples is not added to the state x(k) through a multiplexer as happens for the augmented MPC version but enters in the MPC Controller as different variable.
3.4.2 MPC with Known Disturbance control problem
The idea behind MPC with known disturbance control problem is to perform accurate pre-dictions of the system dynamics including in the prediction the wave contribution knowing its future behaviour. In the previously designed augmented MPC controller it is sufficient to know the current measure τw(k) as it is kept constant for all the prediction horizon, leading to an approximate prediction. In the MPC with known disturbance it is necessary to know also and future measures of induced torque besides the current one, as defined in (3.30), which are used to perform an accurate prediction.
The MPC controller capability to provide an accurate prediction is realized through the adoption of a more sophisticated prediction model, which computes the system dynamics prediction basing on the state x(k), input u(k) and disturbance d(k) dynamics. The prediction model is defined as
x(k + 1) = ADTx(k) + BDTu(k) + BdDT d(k) (3.31)
where
• x(k) = ˙ε(k) ε(k) ˙δ(k) δ(k) ρrv,1(k) ρrv,2(k) ρrv,3(k) ρrv,4(k)T
∈ R8×1 is the state;
• u(k) ∈ R is the manipulable input and corresponds to the command torque Tε(k);
• d(k) ∈ R is the disturbance and corresponds to the measure of τw(k).
The state matrix ADT, the input matrix BDT and the disturbance input matrix BdDT
are obtained from the discretization process already discussed in the augmented MPC control problem in section 3.3, but in the current context these matrices are explicitly used in the model (3.31), and not for developing the augmented model.
3.4.3 Implementation
The implementation of the MPC with known disturbance control problem takes place com-pletely through YALMIP toolbox in this case. This choice is required since the MPT3 toolbox, used for setting up the problem in the other cases, does not allow to set a pre-diction model that involves also a disturbace term besides the states and the input of the system, as the one in (3.31).
Despite YALMIP adopts a different method for defining an MPC control problem, it allows through a series of operation to create an MPC object as well as the MPT3 tool-box does. The created MPC object, thus, gives the the chance to call the optimization function which takes the required data in input and provides the optimal control action Tε(k) as output. This function is called by the MPC Controller shown in figure 3.28 at each sampling time k.
The data required by the optimization function which must be available at each sam-pling time k are the following:
• x(k);
• d(k), d(k + 1), . . . , d(k + Hp).
The optimal tuning of MPC parameters which result to provide the optimal tradeoff maximizing the produced power corresponds to the tuning adopted for the other two control approaches, summarized in table 3.12.
Table 3.12: Parameters of MPC with Known Disturbance Parameter Value Description
Ts 50 ms Sampling Time
Hp 2 Prediction Horizon
q11 1 · 109 Q matrix weight q22 3 · 1010 Q matrix weight
n1 1 N matrix weight
r 0.020 R matrix Weight
Tεmax +100 (kNm) Constraint Tεmin −100 (kNm) Constraint
Chapter 3 - MPC Control for ISWEC 3.4.4 Results
This subsection shows the results obtained by the MPC with known disturbance controller during the simulation of the nonlinear ISWEC model for the nine wave profiles of interest (in table 3.2). Produced power results only are shown as the states dynamics and the command activity result to be analogue to the the standard MPC approach, resulting thus in the same observations.
In figure 3.29 it is reported the absorbed power trend obtained with augmented MPC approach in red and the MPC with known disturbance approach in green.
0 100 200 300 400 500 600 700 800 900 1000
-400 -300 -200 -100 0 100 200 300
MPC AUG MPCDIST
Figure 3.29: Absorbed Power Plot - Augmented MPC vs MPC with Known Disturbance It is possible to observe that the two absorbed power trends result to be identical.
For a closer view it is possible to observe the figure 3.30 which reports a section of the figure 3.29 in which is also illustrated a magnified portion of the plot containing the peak comparison between the augmented MPC and the MPC with known disturbance.
0 20 40 60 80 100 120 140 160
-200 -150 -100 -50 0 50 100
MPCAUG MPCDIST
-116.34 -116.32 -116.30
Figure 3.30: Absorbed Power Plot (section) - Augmented MPC vs MPC with Known Disturbance
From the figures 3.29 and 3.30 it is observable that the two MPC control approaches result to provide the same absorbed power trend.
For a quantitative evaluation the produced mean power has been computed according to the already discussed following formula
P = − 1 ns
ns
X
k = 1
P (k)
Having the values of P it is possible to compute the normalized produced mean power as
P =b P Pmax
where Pmax is the maximum produced mean power among the results obtained by MPC, augmented MPC and MPC with known disturbance.
Results of the normalized produced mean power are reported in table 3.13, where each row reports the wave identification and the normalized produced mean power by the three different MPC approaches. bPM P Cstd , bPM P Caug and bPM P Cdist denote respectively the results obtained with the standard MPC, augmented MPC and the MPC with known disturbance.
Table 3.13: Normalized Produced Mean Power - MPC with Known Disturbance Wave ID PbM P Cstd PbM P Caug PbM P Cdist
1 0.42408 0.42552 0.42552
2 1.00 0.99924 0.99924
3 0.05643 0.05644 0.05644
4 0.08122 0.08125 0.08125
5 0.31046 0.31055 0.31055
6 0.37492 0.37491 0.37491
7 0.16716 0.16717 0.16717
8 0.57373 0.57684 0.57684
9 0.12673 0.12678 0.12678
Also from the quantitative point of view the produced mean power obtained with the MPC with known disturbance approach results to be identical to the one provided by the augmented MPC.
The reason why the produced mean power results to be the same for both the aug-mented MPC and the MPC with known disturbance lies in the prediction horizon Hp. In particular, the tuning of the parameters which allows to obtain the optimal tradeoff between states behaviour, command activity and maximization of the produced power provides for a prediction horizon set to Hp = 2 (see table 3.12), whose value results to be not long enough for allowing differences between the augmented MPC and the MPC with kwown disturbance. Thus, predicting the state evolution of only 2 steps in the future is such that the wave contribution included into the dynamics prediction by the MPC with known disturbance approach has the same effect as the one provided by the augmented MPC approach.
Therefore, in order to observe different results between the MPC with known disturbance and the augmented MPC it is necessary to set the prediction horizon Hp to a value suffi-ciently long. As an example the table 3.14 reports the normalized produced mean power obtained considering the Wave 1 with a prediction horizon set as Hp = 20.
Chapter 3 - MPC Control for ISWEC
Table 3.14: Normalized Produced Mean Power - MPC with Known Disturbance (Hp= 20) Wave ID PbM P Cstd PbM P Caug PbM P Cdist
1 0.990 0.997 1.00
The results reported in table 3.14 show that the MPC with known disturbance is able to produce a greater power than the augmented MPC provided that the prediction horizon is sufficiently long. It is worth to remark that the three normalized produced mean power values in table 3.14 refer to absolute produced mean power values which are lower than the optimal case reported in table 3.13, as in this last case the value of Hp is not the optimal one.
In figure 3.31 it is shown the absorbed power trend by the three different MPC ap-proaches with a prediction horizon set as Hp = 20, when ISWEC nonlinear model is excited by the input wave classified as Wave 1. The absorbed power trends provided by the standard MPC, augmented MPC and the MPC with known disturbance are illustrated by the blue, red and green line respectively.
0 100 200 300 400 500 600 700 800 900 1000
-600 -400 -200 0 200 400 600
MPC MPCAUG MPCDIST
Figure 3.31: Absorbed Power Plot - MPC vs Augmented MPC vs MPC with Known Disturbance (Hp = 20)
The figure 3.32 shows a section of the previous figure 3.31 which shows the different absorbed power capabilities by the three different MPC approaches when the prediction horizon is to Hp = 20.
0 10 20 30 40 50 60 70 80 90 100
-80 -60 -40 -20 0 20 40
MPC MPCAUG MPCDIST
-18.95 -18.90 -18.85 -18.80 -18.75
Figure 3.32: Absorbed Power Plot (section) - MPC vs Augmented MPC vs MPC with Known Disturbance (Hp = 20)
From figure 3.32 it is thus possible to observe that the MPC with known disturbance is able to extract more power than the other two MPC approaches, provided that the prediction horizon is long enough to make differences with respect to the augmented MPC approach.